L10a89

L10a88

L10a90

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Link Presentations

[edit Notes on L10a89's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X18,16,19,15 X16,7,17,8 X6,17,7,18 X8,20,1,19
Gauss code	{1, -5, 2, -6, 4, -9, 8, -10}, {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{(u-1) (v-1) \left(u^2 v^2-u^2 v-u v^2-u-v+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+7 q^{5/2}-7 q^{3/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +5 z a^{-3} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-7 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +4 z^6 a^{-6} -2 z^4 a^{-6} -3 z^2 a^{-6} +4 z^7 a^{-5} -5 z^5 a^{-5} -2 z^3 a^{-5} +2 z a^{-5} +3 z^8 a^{-4} -5 z^6 a^{-4} -z^4 a^{-4} +z^2 a^{-4} +z^9 a^{-3} +3 z^7 a^{-3} -18 z^5 a^{-3} +17 z^3 a^{-3} -4 z a^{-3} +5 z^8 a^{-2} -18 z^6 a^{-2} +15 z^4 a^{-2} -z^2 a^{-2} +z^9 a^{-1} +a z^7-5 a z^5-14 z^5 a^{-1} +8 a z^3+23 z^3 a^{-1} -5 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8-9 z^6+11 z^4-3 z^2-1 }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 2 1   1 10               4 2     -2 8             4 2       2 6           3 4         1 4         4 4           0 2       3 5             2 0     1 2               -1 -2   1 3                 2 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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